C₂²×#45458 #72293730

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁶×C₄²

Aliases

direct product

C₂²×#45458

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

166909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 28 points:

(6,9)(7,11)(13,14)(15,20)(16,18)(17,19)(21,22,25,27)(23,28,26,24);(7,11)(10,12)(13,15,17,16)(14,18,19,20)(21,23)(22,24)(25,26)(27,28);(21,22)(23,28)(24,26)(25,27);(5,6,8,9)(7,12,11,10)(13,16,17,15)(14,18,19,20)(21,22)(23,24)(25,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,20)(17,19)(21,24)(22,26)(23,27)(25,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8598543
maximal normal subgroup of	C₂²×#8598543
maximal quotient of	C₂²×#845298, C₂²×#2032000

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((6,9)(7,11)(13,14)(15,20)(16,18)(17,19)(21,22,25,27)(23,28,26,24), (7,11)(10,12)(13,15,17,16)(14,18,19,20)(21,23)(22,24)(25,26)(27,28), (21,22)(23,28)(24,26)(25,27), (5,6,8,9)(7,12,11,10)(13,16,17,15)(14,18,19,20)(21,22)(23,24)(25,27)(26,28), (5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,20)(17,19)(21,24)(22,26)(23,27)(25,28), (3,4), (1,2)(3,4));

C22×#45458 #72293730